Dynamic service rate control for a single‐server queue with Markov‐modulated arrivals

We consider the problem of service rate control of a single‐server queueing system with a finite‐state Markov‐modulated Poisson arrival process. We show that the optimal service rate is nondecreasing in the number of customers in the system; higher congestion levels warrant higher service rates. On the contrary, however, we show that the optimal service rate is not necessarily monotone in the current arrival rate. If the modulating process satisfies a stochastic monotonicity property, the monotonicity is recovered. We examine several heuristics and show where heuristics are reasonable substitutes for the optimal control. None of the heuristics perform well in all the regimes and the fluctuation rate of the modulating process plays an important role in deciding the right heuristic. Second, we discuss when the Markov‐modulated Poisson process with service rate control can act as a heuristic itself to approximate the control of a system with a periodic nonhomogeneous Poisson arrival process. Not only is the current model of interest in the control of Internet or mobile networks with bursty traffic, but it is also useful in providing a tractable alternative for the control of service centers with nonstationary arrival rates. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 661–677, 2013     

Equilibrium customer strategies and social–profit maximization in the single‐server constant retrial queue

We consider the single‐server constant retrial queue with a Poisson arrival process and exponential service and retrial times. This system has not waiting space, so the customers that find the server busy are forced to abandon the system, but they can leave their contact details. Hence, after a service completion, the server seeks for a customer among those that have unsuccessfully applied for service but left their contact details, at a constant retrial rate. We assume that the arriving customers that find the server busy decide whether to leave their contact details or to balk based on a natural reward‐cost structure, which incorporates their desire for service as well as their unwillingness to wait. We examine the customers' behavior, and we identify the Nash equilibrium joining strategies. We also study the corresponding social and profit maximization problems. We consider separately the observable case where the customers get informed about the number of customers waiting for service and the unobservable case where they do not receive this information. Several extensions of the model are also discussed. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Optimal staffing in nonstationary service centers with constraints

This paper considers optimal staffing in service centers. We construct models for profit and cost centers using dynamic rate queues. To allow for practical optimal controls, we approximate the queueing process using a Gaussian random variable with equal mean and variance. We then appeal to the Pontryagin's maximum principle to derive a closed form square root staffing (SRS) rule for optimal staffing. Unlike most traditional SRS formulas, the main parameter in our formula is not the probability of delay but rather a cost‐to‐benefit ratio that depends on the shadow price. We show that the delay experienced by customers can be interpreted in terms of this ratio. Throughout the article, we provide theoretical support of our analysis and conduct extensive numerical experiments to reinforce our findings. To this end, various scenarios are considered to evaluate the change in the staffing levels as the cost‐to‐benefit ratio changes. We also assess the change in the service grade and the effects of a service‐level agreement constraint. Our analysis indicates that the variation in the ratio of customer abandonment over service rate particularly influences staffing levels and can lead to drastically different policies between profit and cost service centers. Our main contribution is the introduction of new analysis and managerial insights into the nonstationary optimal staffing of service centers, especially when the objective is to maximize profitability. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 615–630, 2017     

Analysis of the finite‐source multiclass priority queue with an unreliable server and setup time

In this article, we study a queueing system serving multiple classes of customers. Each class has a finite‐calling population. The customers are served according to the preemptive‐resume priority policy. We assume general distributions for the service times. For each priority class, we derive the steady‐state system size distributions at departure/arrival and arbitrary time epochs. We introduce the residual augmented process completion times conditioned on the number of customers in the system to obtain the system time distribution. We then extend the model by assuming that the server is subject to operation‐independent failures upon which a repair process with random duration starts immediately. We also demonstrate how setup times, which may be required before resuming interrupted service or picking up a new customer, can be incorporated in the model. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

Scheduling and due‐date quotation in a make‐to‐order supply chain

We consider a manufacturer, served by a single supplier, who has to quote due dates to arriving customers in a make‐to‐order production environment. The manufacturer is penalized for long lead times and for missing due dates. To meet due dates, the manufacturer has to obtain components from a supplier. We model this manufacturer and supplier as a two‐machine flow shop, consider several variations of this problem, and design effective due‐date quotation and scheduling algorithms for centralized and decentralized versions of the model. We perform extensive computational testing to assess the effectiveness of our algorithms and to compare the centralized and decentralized models to quantify the value of centralized control in a make‐to‐order supply chain. Since complete information exchange and centralized control is not always practical or cost‐effective, we explore the value of partial information exchange for this system. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Scheduling policies for an antiterrorist surveillance system

This article concerns scheduling policies in a surveillance system aimed at detecting a terrorist attack in time. Terrorist suspects arriving at a public area are subject to continuous monitoring, while a surveillance team takes their biometric signatures and compares them with records stored in a terrorist database. Because the surveillance team can screen only one terrorist suspect at a time, the team faces a dynamic scheduling problem among the suspects. We build a model consisting of an M/G/1 queue with two types of customers—red and white—to study this problem. Both types of customers are impatient but the reneging time distributions are different. The server only receives a reward by serving a red customer and can use the time a customer has spent in the queue to deduce its likely type. In a few special cases, a simple service rule—such as first‐come‐first‐serve—is optimal. We explain why the problem is in general difficult and we develop a heuristic policy motivated by the fact that terrorist attacks tend to be rare events. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Dynamic capacity reservation and due date quoting in a make‐to‐order system

We consider a make‐to‐order manufacturer facing random demand from two classes of customers. We develop an integrated model for reserving capacity in anticipation of future order arrivals from high priority customers and setting due dates for incoming orders. Our research exhibits two distinct features: (1) we explicitly model the manufacturer's uncertainty about the customers' due date preferences for future orders; and (2) we utilize a service level measure for reserving capacity rather than estimating short and long term implications of due date quoting with a penalty cost function. We identify an interesting effect (“t‐pooling”) that arises when the (partial) knowledge of customer due date preferences is utilized in making capacity reservation and order allocation decisions. We characterize the relationship between the customer due date preferences and the required reservation quantities and show that not considering the t‐pooling effect (as done in traditional capacity and inventory rationing literature) leads to excessive capacity reservations. Numerical analyses are conducted to investigate the behavior and performance of our capacity reservation and due date quoting approach in a dynamic setting with multiple planning horizons and roll‐overs. One interesting and seemingly counterintuitive finding of our analyses is that under certain conditions reserving capacity for high priority customers not only improves high priority fulfillment, but also increases the overall system fill rate. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Price and capacity decisions of service systems with boundedly rational customers

We consider price and capacity decisions for a profit‐maximizing service provider in a single server queueing system, in which customers are boundedly rational and decide whether to join the service according to a multinomial logit model. We find two potential price‐capacity pair solutions for the first‐order condition of the profit‐maximizing problem. Profit is maximized at the solution with a larger capacity, but minimized at the smaller one. We then consider a dynamically adjusting capacity system to mimic a real‐life situation and find that the maximum can be reached only when the initial service rate is larger than a certain threshold; otherwise, the system capacity and demand shrink to zero. We also find that a higher level of customers’ bounded rationality does not necessarily benefit a firm, nor does it necessarily allow service to be sustained. We extend our analysis to a setting in which customers’ bounded rationality level is related to historical demand and find that such a setting makes service easier to sustain. Finally we find that bounded rationality always harms social welfare.     

Rank‐Cluster‐and‐Prune: An algorithm for generating clusters in complex set partitioning problems

Clustering problems are often difficult to solve due to nonlinear cost functions and complicating constraints. Set partitioning formulations can help overcome these challenges, but at the cost of a very large number of variables. Therefore, techniques such as delayed column generation must be used to solve these large integer programs. The underlying pricing problem can suffer from the same challenges (non‐linear cost, complicating constraints) as the original problem, however, making a mathematical programming approach intractable. Motivated by a real‐world problem in printed circuit board (PCB) manufacturing, we develop a search‐based algorithm (Rank‐Cluster‐and‐Prune) as an alternative, present computational results for the PCB problem to demonstrate the tractability of our approach, and identify a broader class of clustering problems for which this approach can be used. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Scheduling customer arrivals to a stochastic service system

An efficient algorithm for determining the optimal arrival schedule for customers in a stochastic service system is developed. All customers arrive exactly when scheduled, and service times are modeled as iid Erlang random variables. Costs are incurred at a fixed rate per unit of time each customer waits for service, and an additional cost is incurred for every unit of time the server operates beyond a scheduled closing time. The objective is to minimize total operating cost. This type of problem arises in many operational contexts including transportation, manufacturing, and appointment‐based services. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 549–559, 1999     

Benders decomposition with alternative multiple cuts for a multi‐product closed‐loop supply chain network design model

In this article, we consider a multi‐product closed‐loop supply chain network design problem where we locate collection centers and remanufacturing facilities while coordinating the forward and reverse flows in the network so as to minimize the processing, transportation, and fixed location costs. The problem of interest is motivated by the practice of an original equipment manufacturer in the automotive industry that provides service parts for vehicle maintenance and repair. We provide an effective problem formulation that is amenable to efficient Benders reformulation and an exact solution approach. More specifically, we develop an efficient dual solution approach to generate strong Benders cuts, and, in addition to the classical single Benders cut approach, we propose three different approaches for adding multiple Benders cuts. These cuts are obtained via dual problem disaggregation based either on the forward and reverse flows, or the products, or both. We present computational results which illustrate the superior performance of the proposed solution methodology with multiple Benders cuts in comparison to the branch‐and‐cut approach as well as the traditional Benders decomposition approach with a single cut. In particular, we observe that the use of multiple Benders cuts generates stronger lower bounds and promotes faster convergence to optimality. We also observe that if the model parameters are such that the different costs are not balanced, but, rather, are biased towards one of the major cost categories (processing, transportation or fixed location costs), the time required to obtain the optimal solution decreases considerably when using the proposed solution methodology as well as the branch‐and‐cut approach. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Approximation results for min‐max path cover problems in vehicle routing

This article studies a min‐max path cover problem, which is to determine a set of paths for k capacitated vehicles to service all the customers in a given weighted graph so that the largest path cost is minimized. The problem has wide applications in vehicle routing, especially when the minimization of the latest service completion time is a critical performance measure. We have analyzed four typical variants of this problem, where the vehicles have either unlimited or limited capacities, and they start from either a given depot or any depot of a given depot set. We have developed approximation algorithms for these four variants, which achieve approximation ratios of max{3 ‐ 2/k,2}, 5, max{5 ‐ 2/k,4}, and 7, respectively. We have also analyzed the approximation hardness of these variants by showing that, unless P = NP , it is impossible for them to achieve approximation ratios less than 4/3, 3/2, 3/2, and 2, respectively. We have further extended the techniques and results developed for this problem to other min‐max vehicle routing problems.© 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

The stochastic single resource service‐provision problem

The service‐provision problem described in this paper comes from an application of distributed processing in telecommunications networks. The objective is to maximize a service provider's profit from offering computational‐based services to customers. The service provider has limited capacity and must choose which of a set of software applications he would like to offer. This can be done dynamically, taking into consideration that demand for the different services is uncertain. The problem is examined in the framework of stochastic integer programming. Approximations and complexity are examined for the case when demand is described by a discrete probability distribution. For the deterministic counterpart, a fully polynomial approximation scheme is known 2 . We show that introduction of stochasticity makes the problem strongly NP‐hard, implying that the existence of such a scheme for the stochastic problem is highly unlikely. For the general case a heuristic with a worst‐case performance ratio that increases in the number of scenarios is presented. Restricting the class of problem instances in a way that many reasonable practical problem instances satisfy allows for the derivation of a heuristic with a constant worst‐case performance ratio. Worst‐case performance analysis of approximation algorithms is classical in the field of combinatorial optimization, but in stochastic programming the authors are not aware of any previous results in this direction. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

Approximation algorithms for general one‐warehouse multi‐retailer systems

Logistical planning problems are complicated in practice because planners have to deal with the challenges of demand planning and supply replenishment, while taking into account the issues of (i) inventory perishability and storage charges, (ii) management of backlog and/or lost sales, and (iii) cost saving opportunities due to economies of scale in order replenishment and transportation. It is therefore not surprising that many logistical planning problems are computationally difficult, and finding a good solution to these problems necessitates the development of many ad hoc algorithmic procedures to address various features of the planning problems. In this article, we identify simple conditions and structural properties associated with these logistical planning problems in which the warehouse is managed as a cross‐docking facility. Despite the nonlinear cost structures in the problems, we show that a solution that is within ε‐optimality can be obtained by solving a related piece‐wise linear concave cost multi‐commodity network flow problem. An immediate consequence of this result is that certain classes of logistical planning problems can be approximated by a factor of (1 + ε) in polynomial time. This significantly improves upon the results found in literature for these classes of problems. We also show that the piece‐wise linear concave cost network flow problem can be approximated to within a logarithmic factor via a large scale linear programming relaxation. We use polymatroidal constraints to capture the piece‐wise concavity feature of the cost functions. This gives rise to a unified and generic LP‐based approach for a large class of complicated logistical planning problems. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Free riding in a multi‐channel supply chain

Free riding in a multichannel supply chain occurs when one retail channel engages in the customer service activities necessary to sell a product, while another channel benefits from those activities by making the final sale. Although free riding is, in general, considered to have a negative impact on supply chain performance, certain recent industry practices suggest an opposite view: a manufacturer may purposely induce free riding by setting up a high‐cost, customer service‐oriented direct store to allow consumers to experience the product, anticipating their purchase at a retail store. This article examines how the free riding phenomenon affects a manufacturer's supply chain structure decision when there are fixed plus incremental variable costs for operating the direct store. We consider factors such as the effort required to find and buy the product at a retail store after visiting the direct store, the existence of competing products in the market, and the extent of consumer need to obtain direct‐store service. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

The effect of catastrophes on the strategic customer behavior in queueing systems

We consider the single server Markovian queue subject to Poisson generated catastrophes. Whenever a catastrophe occurs, all customers are forced to abandon the system, the server is rendered inoperative and an exponential repair time is set on. During the repair time new arrivals are allowed to join the system. We assume that the arriving customers decide whether to join the system or balk, based on a natural linear reward‐cost structure with two types of rewards: A (usual) service reward for those customers that receive service and a (compensation) failure reward for those customers that are forced to abandon the system due to a catastrophe. We study the strategic behavior of the customers regarding balking and derive the corresponding (Nash) equilibrium strategies for the observable and unobservable cases. We show that both types of strategic behavior may be optimal: to avoid the crowd or to follow it. The crucial factor that determines the type of customer behavior is the relative value of the service reward to the failure compensation. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Workload distribution of discrete‐time parallel queues with two servers

We study discrete‐time, parallel queues with two identical servers. Customers arrive randomly at the system and join the queue with the shortest workload that is defined as the total service time required for the server to complete all the customers in the queue. The arrivals are assumed to follow a geometric distribution and the service times are assumed to have a general distribution. It is a no‐jockeying queue. The two‐dimensional state space is truncated into a banded array. The resulting modified queue is studied using the method of probability generating function (pgf) The workload distribution in steady state is obtained in form of pgf. A special case where the service time is a deterministic constant is further investigated. Numerical examples are illustrated. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 440–454, 2000     

Scheduling a production–distribution system to optimize the tradeoff between delivery tardiness and distribution cost

We consider a make‐to‐order production–distribution system with one supplier and one or more customers. A set of orders with due dates needs to be processed by the supplier and delivered to the customers upon completion. The supplier can process one order at a time without preemption. Each customer is at a distinct location and only orders from the same customer can be batched together for delivery. Each delivery shipment has a capacity limit and incurs a distribution cost. The problem is to find a joint schedule of order processing at the supplier and order delivery from the supplier to the customers that optimizes an objective function involving the maximum delivery tardiness and the total distribution cost. We first study the solvability of various cases of the problem by either providing an efficient algorithm or proving the intractability of the problem. We then develop a fast heuristic for the general problem. We show that the heuristic is asymptotically optimal as the number of orders goes to infinity. We also evaluate the performance of the heuristic computationally by using lower bounds obtained by a column generation approach. Our results indicate that the heuristic is capable of generating near optimal solutions quickly. Finally, we study the value of production–distribution integration by comparing our integrated approach with two sequential approaches where scheduling decisions for order processing are made first, followed by order delivery decisions, with no or only partial integration of the two decisions. We show that in many cases, the integrated approach performs significantly better than the sequential approaches. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005